The departure-time choice equilibrium
of the corridor problem
with discrete multiple bottlenecks
modeling, solvability, and uniqueness
Takashi Akamatsu (Tohoku Univ.)
Kentaro Wada (Univ. of Tokyo)
* Shunsuke Hayashi (Tohoku Univ.)
The 27th European Conference on Operational Research (EURO2015), Glasgow, Scotland
Bottleneck and Traffic Congestion
We often encounter the traffic congestion (traffic jam) at the
entrance of a bottleneck (BN).
e.g. bridge, interchange, construction, etc.
The traffic capacity of the BN is limited. There happens the
congestion when the traffic flow concentrates over the BN capacity.
(traffic) flow
BN
flow
flow
congestion!
Economical loss due to the congestion is quite a bit.
It is important to analyze the congestion
due to the bottlenecks.
Existing BN and congestion researches
[Single BN model]
• Definition of departure-time choice equilibrium
• More detailed equilibrium model
[Vickrey 1969]
[Hendrickson et. al., 1981]
• Existence and uniqueness of equilibrium
[Smith, 1984] [Daganzo, 1985]
[Multiple BN model]
• Extension to 2 tandem BNs with equilibrium analyses
[Kuwahara, 1990], [Arnott et al. 1993]
• Extension to N tandem BNs
[Arnott and De Palma, 2011]
Analyses of equilibrium (existence/uniqueness) is not sufficient.
We reformulate the departure time choice equilibrium problem
with N tandem BNs as a linear/nonlinear complementarity problem.
Then, we study its existence and uniqueness property.
Note: This equilibrium model is different from the well-known Wordrop equilibrium,
which is the time-independent equilibrium model based on the route choice.
Corridor Network
Consider many-to-one OD pairs with N tandem bottlenecks.
(morning rush model)
BN N
BN2
BN1
CBD
users
Zone N
・ There exist
・・・
users
users
Zone 2
Zone 1
users (commuters / vehicles) in each Zone .
・ Every morning, all users commute to the same CBD (central business district).
・ Users in Zone
go through the downstream
・ First-In-First-Out (FIFO) is assumed.
Notice the correspondence between zone numbers and BN numbers.
Choice of departure time
defied later
Each user chooses the departure time to minimize his/her travel cost.
： departure time
T is sufficiently large number.
： density of users living in Zone
who choose
： Number of users in Zone
(traffic demand)
Notice: At first, time and number of users are supposed to be continuous.
（We discretize them later.)
This area equals the number
of users departing in
Departure Time Choice Equilibrium (DTCE)
The (traffic) flow pattern satisfying the following 3 conditions
(a)-(c) is called Departure Time Choice Equilibrium (DTCE).
(a) Queuing Condition
How is the queue generated at BN? How long do we wait at the queue?
(b) Departure Time Choice Policy
How does each user choose his/her departure time?
(c) Flow Conservation Law
BN
(a) Queuing Condition
(single BN case)
: time
: cumulative num. of users
arriving at the entrance of BN
gradient =
(capacity of BN)
: cumulative num. of users
departing from BN
: capacity of BN
time
(maximum number of users who can
pass through the BN per unit time)
： num. of users queuing at BN
: waiting time in the queue
Note: Physical size of queue
or vehicles are ignored.
：departure rate
(a) Queuing Condition
(single BN case)
&
The waiting time
has to satisfy the above complementarity condition.
We want to extend this formulation to the multiple BN model.
Extention to multiple BN case
(based on the existing time coordinate system)
If we directly inherit the time coordinate system from the single BN case,
then the time variables becomes quite nested.
impossible to analyze!
In stead of using actual time t, we introduce
a new time coordinate system based on CBD arrival time s.
Time coordinate
based
on
The usersystem
chooses his/her
departure
timeCBD arrival
so that the CBD arrival time is s.
: arrival time at CBD
sufficiently large interval
: arrival time at the entrance of
(of the user whose CBD arrival time is s)
(Due to FIFO,
do not depend on the zone in which user lives.)
Single BN case
: cumulative num. of users
arriving at the entrance of BN
: waiting time in the queue
Multiple BNs (complementarity
condition for
Ay
z
Note: The capacity of each BN can be different.
)
Time coordinate system based on CBD arrival
: waiting time at
(w.r.t. new coordinate system s)
: cumulative number of users arriving at
(w.r.t. new coordinate system s)
We have derived the queuing condition based on the CBD arrival time s.
Departure Time Choice Equilibrium (DTCE)
The (traffic) flow pattern satisfying the following 3 conditions
(a)-(c) is called Departure Time Choice Equilibrium (DTCE).
(a) Queuing Condition
(b) Departure Time Choice Policy
How does each user choose his/her departure time?
(c) Flow Conservation Law
We say “departure time choice” in the sense that each driver chooses
the departure time so that the CBD arrival time becomes s.
Users’ Time Choice
Each user chooses CBD arrival time
： CBD arrival time
： density of users living in Zone
who choose
: number of users in Zone
This area equals the number
of users in Zone arriving at
CBD in the interval
Users’ Time Choice
User in Zone
chooses his/her CBD arrival time to minimize the cost function.
: arrival time at
schedule delay penalty
late arrival
penalty
: desired arrival time
early arrival
penalty
: desired arrival time
In this study, we assume that all users
in every zone have the same schedule
delay penalty function.
Time Choice Policy A (min type)
[Policy A] min type (deterministic time choice)
Time Choice Policy B (logit type)
[Policy B] logit type (probabilistic time choice)
Time Choice Policy B (logit type)
[Policy B] logit type (probabilistic time choice)
Density of users
Note: When
choosing CBD arrival time
, the logit type approaches to the min type.
Departure Time Choice Equilibrium (DTCE)
The (traffic) flow pattern satisfying the following 3 conditions
(a)-(c) is called Departure Time Choice Equilibrium (DTCE).
(a) Queuing Condition
(b) Departure Time Choice Policy
(min type)
or
(logit type)
(c) Flow Conservation Law
(c) Flow Conservation Law
When the time choice policy is logit type, this condition
holds automatically.
logit case
Departure Time Choice Equilibrium (DTCE)
(a) Queuing Condition
(b) Departure Time Choice Policy
(min type)
or
(logit type)
(c) Flow Conservation Law
(min type)
Infinite dimensional complementarity reformulation
Combining three conditions (a)-(c), we obtain the following infinite
dimensional complementarity reformulation
min type
infinite dimensional LCP
logit type
infinite dimensional NCP
Note: Since
and
,
,
, we eliminated those variables.
Finite LCP reformulation by time descretization
min type
Time descretization:
(K: descretization number)
where
Finite LCP reformulation by time descretization
logit type
Time descretization:
(K: descretization number)
Evening rush model (one-to-many)
We can also consider the evening-rush model.
BN N
BN2
BN1
CBD
commuters
Zone N
・・・
commuters
Zone 2
commuters
Zone 1
Basically, the complementarity model can be obtained by
re-defining the time coordinate system in a backward direction.
Evening rush model (one-to-many)
Evening rush model (min type)
Evening rush model (logit type)
Existence of equilibrium
For each model, we have shown the existence of
the departure time choice equilibrium.
Theorem
There exist at least one equilibrium in either case:
(i) Morning rush model (min type)
(ii) Morning rush model (logit type)
(iii) Evening rush model (min type)
(iv) Evening rush model (logit type)
Overview of the proof (morning/min case)
(a)
(b)
(c)
In (a), w can be expressed as a function of q.
In (b) and (c), q is expressed as a set-valued mapping of w.
If the fixed point problem
has a solution, then the above LCP also has a solution.
is upper semi-continuous, and any image is nonempty, closed and convex.
- The domain of
is nonempty, compact, and satisfies
We can apply Kakutani’s fixed-point theorem.
Overview of the proof (other cases)
(Evening rush model: min type)
Proved by
Kakutani’s theorem
(Morning rush model: logit type)
(Evening rush model: logit type)
Proved by
Brower’s theorem
Uniqueness of equilibrium
Evening rush model (logit type)
Proved.
(We analyzed the determinant of Jacobian appearing
in the NCP reformulation.)
Morning rush model (logit type)
Proved under a certain assumption.
Evening rush model (min type)
Morning rush model (min type)
Not Proved.
• We have confirmed experimentally that the
uniqueness is satisfied with respect to w.
• With respect to q, we have a counterexample
such that the uniqueness does not hold.
Numerical example (morning rush model: logit type)
#BN：N=3, time #disc.: K=60, BN capa：μ = (30,20,10), # users Q=(100,200,300)
Red: cum. arrival to BN,
BN3 (upstream)
Blue: cum. departure from BN
BN2 (middle stream)
BN1(downstream)
Total of
all zones
Zone 1
residents
Zone 2
residents
Zone 3
residents
Numerical example (evening rush model: logit type)
#BN：N=3, time #disc.: K=60, BN capa：μ = (30,20,10), # users Q=(100,200,300)
Red: cum. arrival to BN,
BN3 (upstream)
Blue: cum. departure from BN
BN2 (middle stream)
BN1(downstream)
Total of
all zones
Zone 1
residents
Zone 2
residents
Zone 3
residents
(Final Slide) Information of conference
Registration Deadline: Jul 22, 2015
• ISTTT is the top conference in the area of transportation theory.
• This talk will be also presented by my co-author, and published in the
Special Issue of Transportation Research Part B: Methodologial.